For downloading large video files from a server, is it safer (meaning less chance of error and better at preserving file integrity) to do a one-part download like how FileZilla does it? Or is it completely safe to use CuteFTP and to utilize its multi-party downloading feature? If so, what is a good number of threads to open at the same time?

It's more of a issue of how many threads/connections the server allows you to open. I don't see any issue with doing multi-part downloads as basically any download manager out there (not limited to FTP) does this as well.